Analytic approach to Yor’s formula of exponential additive functionals of Brownian motion

Abstract

Yor [4] obtained an exact formula for a one-dimensional Brownian motion {B _t }:

$${{E}_{0}}(f(\int_{0}^{t}{{{e}^{2{{B}_{S}}}}d}s)g({{e}^{{{B}_{t}}}}))=c(t)\int_{0}^{\infty }{dzg(y)f(\frac{1}{z})}\exp \left\{ -\frac{z(1+{{y}^{2}})}{2}{{\psi }_{yz}}(t) \right\},$$

$$c(t)={{(2{{\pi }^{3}}t)}^{-\frac{1}{2}}}\exp (\frac{{{\pi }^{2}}}{2t}),{{\psi }_{r}}(t)=\int_{0}^{\infty }{\exp }(-\frac{{{u}^{2}}}{2t}-\gamma \cosh u)\sinh u\sin (\frac{\pi u}{t})du.$$